• It is tempting to use simulations – particularly computer based
simulations – to help students "discover" a concept, or to help them
understand a concept. This is trickier than it seems. If you are not
careful, students will see the simulation as a black box, so the
following five-step process might be useful whenever students are
working with simulations.

First, answer: Why are we conducting the
simulation? What question are we investigating? What do we hope to
learn?

Then determine: On what assumptions is the
simulation based? E.g. there are two equally likely outcomes, the
population proportion is 60%, etc.

Design and conduct the simulation with a “hands-on”
experience with the random experiment, for example using a die or a
deck of cards. Explain how the assumptions are reflected in the design
of the simulation, e.g. heads means a girl, digits 0-5 mean a “yes”,
etc. Practice a few simulated trials together, interpret, and plot the
results.

Predict what will happen.

Transition to technology going through the same
steps. Explain how the assumptions are reflected in the design of the
simulation, run a few trials, interpret and plot.

• Students will not learn by running other people's simulation
programs. They must do their own simulations.

• It's very important for students to understand exactly what is
happening at each step of a simulation exercise.

• Students must do many different simulations before the idea "sinks"
in.

• Students should be able to do a simulation using a random number
table AND using a random number generator on a calculator or computer.

• There is no rule for deciding when your approximation of the
theoretical probability is "good enough." "Good enough" depends on the
context and the accuracy required of the context.

Student
Misconceptions and Confusions

• Beware: simulations can be intricate and students tend to lose sight
of the goal. It's about estimating a probability (or modeling the
distribution of probabilities), not performing an intricate series of
taps on a calculator.

• Don't confuse lower case n -- the number of objects in a sample in a
single trial -- with N, the number of times you repeat the trial in
your simulation. For example, a trial might consist of selecting 100
Californians at random and counting the number who voted for Arnold
Schwarzenegger, under the assumption that 60% of the population
supported him. To simulate these, we have a bag of six red chips and
four black chips. We draw 100 chips with replacement and count the
number of reds. We repeat this 5000 times. In this example, n = 100 and
N = 5000. (No one said simulations were easy!)

• Some students may wonder what it means for trials to be independent.
In the context of a simulation, independent trials means the outcome of
any one trial does not affect our assessment of the probability
distribution for the outcomes of any other trials.

Resources

• Galton built a famous mechanical simulation: the quincunx. This
simulation illustrates the Central Limit Theorem for the binomial
distribution and could also be used to calculate approximate
probabilities involving adding a series of yes/no random outcomes. You
can view a computer version of it here: RAND Central
Limit Theorem in Action. There are many other quincunx
displays, but this is the best.

• A classic probability problem that can be fun to simulate with your
students is the birthday problem (assuming you have over 30 students in
your class.) One possible moral of this problem is that coincidences
are more likely than one might think. Which leads us to think that
sometimes what we perceive as meaningful patterns are actually simply
due to chance. Here is an applet that demonstrates this problem:
http://www-stat.stanford.edu/~susan/surprise/Birthday.html